Abstract
If L is a star body in R n whose central (n-i)-slices have the same (n-i)-dimensional measure μ n-i (with appropriate density) as the central (n-i)-slices of an origin-symmetric star body K, then the corresponding n-dimensional measures μ n of K and L satisfy μ n (K) ≤ μ n (L). This extends a generalized Funk’s section theorem for volumes to that for measures.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Lutwak E. Intersection bodies and dual mixed volumes [J]. Advances in Mathematics, 1988, 71: 232–261.
Busemann H, Petty C M. Problems on convex bodies [J]. Mathematica Scandinavica, 1956, 4: 88–94.
Gardner R J. A positive answer to the Busemann-Petty problem in three dimensions[J]. Annals of Mathematics, 1994, 140: 435–447.
Koldobsky A. Fourier Analysis in Convex Geometry [M]. Providence: AMS, 2005.
Zhang G. A positive answer to the Busemann-Petty problem in R4 [J]. Annals of Mathematics, 1999, 149: 535–543.
Funk P. Uber Flachen mit lauter geschlossenen geodatischen Linien [J]. Mathematische Annalen, 1913, 74: 278–300.
Gardner R J. Geometric Tomography [M]. New York: Cambridge University Press, 1995.
Zvavitch A. Gaussian measure of sections of convex bodies [J]. Advances in Mathematics, 2004, 188: 124–136.
Zvavitch A. The Busemann-Petty problem of arbitrary measures [J]. MathematischeAnnalen, 2005, 331: 867–887.
Lv S J, Leng G S. Cross i-sections of star bodies and dual mixed volumes [J]. Proceedings of American Mathematical Society, 2007, 135: 3367–3373.
Helgason S. Groups and Geometric Analysis [M]. New York: Academic Press, 1984.
Helgason S. The Radon Transform [M]. Boston: Birkhauser, 1999.
Rubin B. Inversion formulas for the spherical Radon transform and the generalized cosine transform [J]. Advances in Applied Mathematics, 2002, 29: 471–497.
Grinberg E, Zhang G. Convolutions, transforms and convex bodies [J]. Proceedings of London Mathematical Society, 1999, 78: 77–115.
Zhang G. Sections of convex bodies [J]. American Journal of Mathematics, 1996, 118: 319–340.
Milman E. Generalized intersection bodies [J]. Journal of Functional Analysis, 2005, 240: 530–567.
Lv S J. On an analytic generalization of the Busemann-Petty problem [J]. Journal of Mathematical Analysis and Applications, 2008, 341(2): 1438–1444.
Lü S J, Leng G S. On the generalized Busemann-Petty problem [J]. Science in China, Series A, 2007, 50: 1103–1116.
Zhang G. Centered bodies and dual mixed volumes [J]. Transactions of American Mathematical Society, 1994, 345: 777–801.
Author information
Authors and Affiliations
Corresponding author
Additional information
Foundation item: Supported by the National Natural Science Foundation of China (10801140), Chongqing Research Program of Basic Research and Frontier Technology (2013-JCYJ-A00005), and the Foundation of Chongqing Normal University ( 13XLZ05)
Biography: LÜ Songjun, male, Professor, research direction: convex geometric analysis.
Rights and permissions
About this article
Cite this article
Lü, S. A generalized funk’s section theorem for measures. Wuhan Univ. J. Nat. Sci. 22, 313–317 (2017). https://doi.org/10.1007/s11859-017-1252-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11859-017-1252-3